SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

216 P. van Nieuwenhuizen, SupergravityClearly, only the tensors with O~= y, survive since i 10~ is~only antisymmetric for 0~= y~ or ~7sO,.~7s= oP.,.,,. One findsbut[S( ~),S( 2)]A=~.(E2y~ 1)O,.A—~(i2y’ 1)(-yJA). (9)Thus, only on-shell where IA =0, does one find the same result as for A and B.In principle one could define a new symmetry S’A = ~‘yJA, 0’A = S’B =0. This would not besufficient to close the algebra, since new commutators involving 8’ will lead to new symmetries 0”, andso on. The only way to obtain a finite-dimensional closed algebra is to introduce F and G. Indeed, theircontribution to (8) is~(iJA) 2—~(i1ysIA)ys 2— 1 ~-*2 (10)and after a Fierz transformation, the IA terms in eqs. (10) and (9) cancel.For any set of fields and set of global symmetry operations which closes on-shell, one can try to find aset of extra fields which lead to closure off-shell. 2 theories) In general, they these propagate. new fields Therewill is some be nonpropagating confusion the inthe literature action, asbut to sometimes whether auxiliary (for example fields in canR have derivatives in the action, and can be gauge fields. Asimple example due to Ogievetski and Sokatchev suffices to clear this point up. Consider [347]2’ = 1.””T,.~O,,A,,. (11)Clearly, the A,, field equation is 1.””8~T,.~ = 0 and can be solved T,.~= t9,.t,, — 3~t,..Similarly, the T,.,,field equation yields A,, = O,,A. Also T,.,. and A,, are gauge fields since the action is invariant under0T,.~= O,.A,, — O,,A,. or SA,, = O,,A. In a Hamiltonian formalism, the moments for A0 and T,.() lead toso-called primary constraints (ph,, — “’“ T,.0 = 0) and this shows that there 2 poles. are noHence dynamical theremodesis noassociated propagationwith between A,~and the sources. T,.0. Also Thus, in the auxiliary propagators fields can onebefindsgaugeno fields k and carry derivatives in theaction.In order that the reader may test his understanding, we close this section with a second model ofglobal supersymmetry: the photon—neutrino system. The action is2’ = —-~F,.~2 — ~AIA+ ~D2, F,.~= O,.B,, — O~B,. (12)and is invariant under SB,. = —~iy,.A, SA = ~o~”F,.,+ ~iy5D , SD = ~iëy~IA.In this case the countingshows that there are four fermionic components A” and only three photon components, hence the needfor the single auxiliary field D. The reason that there are only three components of B,. is that there is agauge invariance in the algebra as follows from the commutator[S~( ,), 8o( 2)]41 = ~~27~ ! O,.4’ + Sgaugc(11 =Sgauge(11)B,. = 3,.A, Ogauge(1t )A = Sgauge(A )D = 0.(13)In general there are as many field components absent as there are local gauge parameters. This is ofimportance for supergravity, to which we now turn.